We consider Borel sets of finite rank A C A'" where cardinality of A is less than some uncountable regular cardinal K5. We obtain a "normal form" of A, by finding a Borel set Q, such that A and Q continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum. to multiplication by a countable ordinal, and to ordinal exponentiati...

The paper is devoted to the study of topologies on the group Aut(X, B) of all Borel automorphisms of a standard Borel space (X, B). Several topologies are introduced and all possible relations between them are found. One of these topologies, τ , is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of Aut(X, B) and find their closures. ...

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models ...

We study some properties of smooth Borel sets with respect to a Borel equivalence relation, showing some analogies with the collection of countable sets from a descriptive set theoretic point of view. We found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets. We generalize a theorem of Weiss from Z-actions to actions by arbitrary countable groups. We sho...

The optimal conformal mapping of the Borel plane was recently used to accelerate the convergence of the perturbation expansions in QCD. In this work we discuss the relevance of the method for the calculation of the Laplace-Borel integral expressing formally the QCD Green functions. We define an optimal expansion of the Laplace-Borel integral in the principal value prescription and establish con...

For f : [0, 1] → R, consider the relation Ef on [0, 1] defined by (xn)Ef (yn)⇔ P n<ω f(|yn−xn|) <∞. We study the Borel reducibility of Borel equivalence relations of the form Ef . Our results indicate that for every 1 ≤ p < q < ∞, the order ≤B of Borel reducibility on the set of equivalence relations {E : EIdp ≤B E ≤B EIdq} is more complicated than expected, e.g. consistently every linear order...

Harvey Friedman has shown the following completeness theorem: Theorem 0.1 (Friedman) Any countable consistent theory has a Borel model of size continuum. We show a kind of incompleteness theorem at the Borel level: Theorem 0.2 There is a complete, consistent Borel theory of size continuum with no Borel model. The proof works in stages: First we construct a Borel theory T; then we show that all ...

We prove that the equivalence of pure states of a separable C*-algebra is either smooth or it continuously reduces [0, 1]N/l2 and it therefore cannot be classified by countable structures. The latter was independently proved by Kerr–Li–Pichot by using different methods. We also give some remarks on a 1967 problem of Dixmier. If E and F are Borel equivalence relations on Polish spaces X and Y , ...

The following two theorems give the flavour of what will be proved. THEOREM. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0, 1] to Y coincide if and only if Y is connected and locally connected. THEOREM. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class fun...